Actually, Kiyosi Ito solved the measure problem for the Feynman integral in 1960 (see Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume II, U. Cal Press, 1961. pp 227-238) He solves the problem for a non-relativistic H for a free particle and for a particle in a constant force field. He also solves the measure problem for the Weiner Integral, essentiall the Feynman integral after a Wick rotation ( t-> it), which describes brownian motion/heat flow.

The idea is to build a sequence of probability densities (measures) for absolutely continuous trajectories, x(t), and take the appropriate limits. Very heavy math.

Ito also points out that M. Kac, and Gelfand and Yaglom had worked out rigorous approaches the Feynman's path integral. I would suspect that more has been done since that time.

The most successful effort is detailed in glimm's big book wherein he converts the integrals into Wiener interals (which are properly defined) by means of what a physicist would call a"wick rotation" as mentioned above. My understading is that "cameron's thm" shows that there are no appropriate measures in the general case.